Simplify and expand the following expression: $ \dfrac{2q}{q + 1}+\dfrac{-3}{4q - 4} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(q + 1)(4q - 4)$ Multiply the first term by $\dfrac{4q - 4}{4q - 4}$ $ \begin{align*} \dfrac{2q}{q + 1} \times \dfrac{4q - 4}{4q - 4} & = \dfrac{(2q)(4q - 4)}{(q + 1)(4q - 4)} \\ & = \dfrac{8q^2 - 8q}{(q + 1)(4q - 4)}\end{align*} $ Multiply the second term by $\dfrac{q + 1}{q + 1}$ $ \begin{align*} \dfrac{-3}{4q - 4} \times \dfrac{q + 1}{q + 1} & = \dfrac{(-3)(q + 1)}{(4q - 4)(q + 1)} \\ & = \dfrac{-3q - 3}{(4q - 4)(q + 1)}\end{align*} $ Now we have: $ = \dfrac{8q^2 - 8q}{(q + 1)(4q - 4)} + \dfrac{-3q - 3}{(4q - 4)(q + 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{8q^2 - 8q - 3q - 3}{(q + 1)(4q - 4)} $ $ = \dfrac{8q^2 - 11q - 3}{(q + 1)(4q - 4)}$ Expand the denominator: $ = \dfrac{8q^2 - 11q - 3}{4q^2 - 4}$